† 24. By adding and subtracting the expressions found in Art 20, we have R sin (A+B)+R sin (A−B)=2 sin A cos B; If we write S in place of the sum A+B, D in place of the difference A-B, S and D will be two angles of which S is the greater; and A= (S+D), B=1⁄2 (S—D) (Lemma); so that the above expressions become (S-D); R (sin S+sin D)=2 sin (S+D) cos R (sin S-sin D)=2 cos(S+D) sin R (cos S+cos D)=2 cos (S+D) cos R (cos D- -cos S)-2 sin (S+D) sin which are of considerable utility in effecting reductions in the higher branches of the mathematics. † 25. COR. Sin S+sin (S-D): cos(S+D) sin But sin (S+D): cos cos (S-D): sin therefore (VI. L), sin D: sin S-sin D:: sin (S+D) cos (S+D): : tan (S+D): R, (S-D): : tan (S+D): tan (S-D); consequently, sin S+sin D: sin S-sin D: : tan (S+D): tan (S-D). Similarly cos D-cos S: cos D+cos S :: tan } (S+D) . tan } (S-D) : R2. † 26. PROP. 2. Given the tangents of A and B, to find those of A+B and A-B, when the radius is unity. 28. EXAMPLE 1. To find the sine and cosine of 75°. Since 75°-45°+30°, sin 75°=sin 45° cos 30°+cos 45° sin 30° (Art. Again, cos 75° = cos 45° cos 30°- sin 45° sin 30° 30. Ex. 2. To find the cosines of 710, 330, &c. Since cos 2A= 2 cos 2A-1, cos 2A 1+ cos 2A 2 .991445 nearly. So cos 330 = + cos 71° = .997859 nearly. 31. PROP. III. To find approximately the ratio of the circumference of a circle to its diameter. Let polygons of 48 equal sides be inscribed in and described about the circle; and let the radius of the circle be unity. Then the perimeter of the inscribed polygon is 48 chord 710, and, consequently (Art. 6), the semi-perimeter is 48 sin 330. In like manner, the semi-perimeter of the polygon described about the circle is 48 tan. 330. But the semi-circumference of the circle is greater than one of these, and less than the other (I. Sup. Ax. 1). Now 48 sin 33° = 48 1 cos 2330 (Art. 13 (6)) √2304 — (48 × .997859) 2 (Ex. 2) = 3.14 nearly: and tan 330 sin 330 = 3.14....; hence we conclude that 3.14 is a very close cos 320 approximation to the semi-circumference of a circle whose radius is unity. Also, it has been proved (Art. 12) that, in circles of = 2 = different radii, the sides of an inscribed or circumscribed polygon of a given number of sides are as the radii; hence the semi-circumference of a circle whose radius is unity is the ratio of the semi-circumference to the radius, whatever the radius may be; or, which is the same thing, 3.14 is a near approximation to the ratio of the circumference of a circle to its diameter. By carrying the process still farther, we obtain 3.1416; an approximation sufficiently close for practical purposes. 32. PROP. IV. Construction of Trigonometrical Tables. The processes which we have given serve for the determination of the arithmetical values of the trigonometrical functions of a large number of arcs. For, if we know the value of one such function, we can, by constantly halving it, determine those of others. In this way, the cosine of 1°, 52', 30", and its successive halves may be determined. Thus, after twelve bisections of the arc of 60°, the cosine of 52" 44"", 3"", 45"""' is found; and thence also the sine of the same arc. But it is manifest that the sines of very small arcs are to one another nearly as the arcs themselves. For it has been shown (I. Sup. 3) that the number of the sides of an equilateral polygon inscribed in a circle may be so great, that the perimeter of the polygon and the circumference of the circle may differ by a line less than any given line, or, which is the same thing, may be nearly to one another in the ratio of equality. Therefore, their like parts will also be nearly in the ratio of equality, so that the side of the polygon will be to the arc which it subtends nearly in the ratio of equality; and, therefore, half the side of the polygon to half the arc subtended by it, that is to say, the sine of any very small arc will be to the arc itself nearly in the ratio of equality. Hence, we shall have sin l': sin 52", 44", 3"", 45""":: 256: 225, from which the sine of 1' becomes known. It is found to be = .000,2908882. The sine of 1' being found, the sines of 2', 3', or of any number of minutes, are found by Art. 21 ; and their cosines by Art. 13 (6); thence their tangents, by (1), their secants by (2), their cotangents by (3) or (4), and their cosecants by (5) of the same article. Moreover, as we know by other methods the sines and cosines of certain arcs, we may either use these as starting-points from which to determine the values of others, or may proceed in a series of calculations from other commencements until we arrive at these, in which their values, independently obtained, furnish us with the means of verifying the accuracy of our operations. SECTION III. PROPERTIES OF TRIANGLES. 33. PROP. I. In a right-angled plane triangle, as the hypotenuse to either of the sides, so is the radius to the sine of the angle opposite to that side; and as either of the sides is to the other side, so is the radius to the tangent of the angle opposite to the latter. Let ABC be a right-angled plane triangle, of which BC is the hypotenuse. From the centre C, with any radius CD, describe the arc DE; draw DF at right angles to CE, and from E draw EG touching the circle in E, and meeting CB in G; DF is the sine, and EĞ the tangent of the arc DE, or of the angle C. B The two triangles DFC, BAC are equiangular, because the angles DFC, BAC are right angles, and the angle at C is common. Therefore, CB: BA :: CD: DF; but CD is the radius, and DF the sine of the angle C (Def. 4); therefore CB : BAR : sin C. Also, because EG touches the circle in E, CEG is a right angle, and therefore equal to the angle BAC; and since the angle at C is common to the triangles CBA, CGE, these triangles FE A are equiangular, wherefore, CA: AB:: CE: EG; but CE is the radius, and EG the tangent of the angle C; therefore, CA: AB:: R: tan C. 34. COR. 1. As the radius to the secant of the angle C, so is the side adjacent to that angle to the hypotenuse. For CG is the secant of the angle C (Def. 7), and the triangles CGE, CBA being equiangular, CA: CB:: CE: CG, that is, CA: CB:: R: sec C. 35. COR. 2. As either side is to the other, cotangent of the angle opposite to the former. plement of C; and, therefore, tan C=cot B. 36. SCHOLIUM. so is radius to the For B is the com The proposition just demonstrated is most easily remembered, by stating it thus:-If in a right-angled triangle the hypotenuse be made the radius, the sides become the sines of the opposite angles; and if one of the sides be made the radius, the other side becomes the tangent, and the hypotenuse the secant of the opposite angle. 37. PROP. II. The sides of a plane triangle are to one another as the sines of the opposite angles. A From A any angle in the triangle ABC, let AD be drawn perpendicular to BC. And because the triangle ABD is right angled at D, AB: AD:: R:sin B; and, for the same reason, AC: AD :: R: sin C, and, inversely, AD: AC:: sin C: R; therefore, ex æquo inversely, AB: AC :: sin C sin B. In the same manner, it may be demonstrated that AB: BC:: sin C: sin A. Therefore, &c. Q. E. D. B D C COR. If A be a right angle, sin A=R (Art. 5); therefore, CB : BAR sin C, which was proved in Prop. I. 38. PROP. III. In any triangle, twice the rectangle contained by any two sides is to the difference between the sum of the squares of those sides and the square of the base, as the radius to the cosine of the angle included by the two sides. Let ABC be any triangle, 2AB.BC is to the difference between AB2+BC2 and AC2 as radius to cos B. From A draw AD perpendicular to BC, and (II. 12 and 13) the difference between the sum of the squares of AB and BC and the square of AC is equal to 2BC.BD. But BC.BA: BC.BD :: BA: BD :: R: cos B; therefore, also, 2BC.BA: 2BC.BD B A D C :: R: cos B. Now 2BC.BD is the difference between AB2+ † But if B be an obtuse angle, BD is no longer to BA as R to cos B, but as R to cos (180-B), which (Art. 19) is the same thing in value but different in sign. Also, in this case, the square of AC is greater than the sum of the squares of AB,BC. Hence both sides of the above equality are negative in sign. But as the negatives of equal things are equal, the same expression will re ' |